Some critical point theorems and applications to semilinear elliptic partial differential equations

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

P ^AUL H. R ^ABINOWITZ

Some critical point theorems and applications to semilinear elliptic partial differential equations

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 5, n

^o

1 (1978), p. 215-223

<http://www.numdam.org/item?id=ASNSP_1978_4_5_1_215_0>

© Scuola Normale Superiore, Pisa, 1978, tous droits réservés.

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale.

Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

Applications

to

Semilinear Elliptic Partial Differential Equations (*).

PAUL H.

RABINOWITZ (**)

dedicated to Jean

Leray

.Introduction..

Let E be a real Banach space and I ^a

continuously

differentiable _map from E to

R,

^{i.e. I E}

C1(E, R).

We say I satisfies the Palais-Smale condi- tion

(PS)

if any sequence

(um)

^{such that}

l(um)

^{is bounded and - 0}

is

precompact.

^In

[1]

^and

[2], by imposing

^various

qualitative

^conditions

on I near 0 and _oo, Ambrosetti and the author obtained several results con-

cerning

^the existence of critical

points

^{of I and}

applied

^{them to semilinear}

elliptic boundary

^value

problems

^to

get

^{existence theorems in that}

setting.

The purpose of this paper is ^to extend the

theory

^of

[1]

^and

[2].

^{To be}

more

precise,

^let

Br

⁼

r}.

^{It was shown in}

^[1,

^Theorem

^2.1]

that if

I ( o )

^{= 0 and}

1)

^{There are}

constants e,

^a &#x3E; 0 such that I &#x3E; 0 in

Ben(0)

^and

on and

2)

^{There is an e}

E E, e =1=

^{0 such that}

I(e)

⁼

0,

then I has a critical value c &#x3E; a.

It was further shown in

[1,

^Theorem

2.21], [2,

^Theorem

3.37]

^{that if I}

is even and

2)

^is

replaced by

^the

requirement

^{that I be}

negative

^at

infinity

(*) Sponsored by

the United States

Army

under Contract No. DAAG29-75- C-0024 and ,the ^Office of Naval Research under Contract No. N00014-76-C-0300.

Reproduction

^{in whole or in}

part

^is

permitted

^for any purpose of the United States Government.

(**)

Mathematics

Department

and Mathematics Research Center,

University

of Wisconsin, ^{Madison 53706.}

Pervenuto alla Redazione il 4

Giugno

^1977.

216

in an

appropriate

^sense

(see

^condition

^{(I,) of §} 1),

^then

1)

^can be weakened to hold in

Be

^r)

E

where

E

is a

subspace

of E of finite codimension. More-

over for this case I has an unbounded sequence of

positive

critical values.

Our main results here show how ^{one can} still obtain critical

points

^{of I} under weakened versions of

1)

^without

requiring

^{that I be even. The ab-}

stract critical

point

theorems will be

given in § 1

^{and some}

applications

^to semilinear

elliptic partial

differential

equations

^{will be carried out}

in §

^2.

1. - The critical

point

^theorems.

In this section we shall extend the results of

[1]

^and

[2]

^{mentioned in}

the Introduction. Some additional variants ^will be

presented

^{for the}

special

case of E ⁼ Rn.

Lastly

^{a case where we}

only require 1)

^for

Be

intersected with a finite dimensional

subspace

of .E will be treated. As in

[1], [2],

^our

arguments

^{are based on} minimax characterizations of critical values of I.

Our main result is

THEOREM 1.1..Let E be a real Banach space and let ^{I E}

C-I(E, R)

^and

satisfy (PS). Swppose

^{that E =}

Ek

EB

L

where

Ek

^{is k} dimensional and I

satisfies

(11)

(7~)

^{There are constants} e, cx &#x3E; 0 such that

1&#x3E;0

ⁱⁿ

Be

ⁿ

Ê

and

I ~ a

^on

n

Ê.

(I,)

^{For each}

finite

dimensional

subspace R of E,

^{there is a constant R}

= R(R)

such that

I c

^{0 on}

E""-.B R(É) .

^..

Then I has a

positive

^critical

value,

c, characterized

by

where =-:

Ek

EÐ span

for

^some

fixed

T ^E

~B~0},

^{r = and}

A finite dimensional version of Theorem 1.1 was

given

ⁱⁿ

[5]

^{and it}

motivated this paper. Two

preliminary

^{results are}

required

^{for the}

proof

of Theorem 1.1. The

iirst

is a standard lemma from the calculus of variations.

Let

AS

^{= and}

.Ks

^{= = sand}

I’(u)

⁼

0}.

LEMMA 1.3. Let I E

C’(E, R)

^and

satisfy (PS).

^{Let c E}

R,

c~ be any

neigh-

borhood

of K,,

^{0. Then there is an 8 E}

(0, ë)

and an ?7 ^E

C([0, ^1] XE, E)

such that

See e.g.

[2]

^or

[3]

^{for a}

proof

^of this lemma. Next we need a

topological

lemma. and

the

orthogonal complement

^{of R" in Rm.}

LEMMA 1.4. Let and let

If

~o C .R,

^and there is a

homotopy

n such that

G(0, x)

^{= x and}

x)

⁼

g(x),

^{then n}

n

aB~

ⁿ

(Rk ) 1 ~ ^0.

PROOF. A

proof

of Lemma 1.4 due to E. Fadell

using

intersection the- ory

[4,

p.

197]

^{can be found in}

[5,

^Lemma

A-2].

^For the convenience of the

reader,

^{we include it} here. The

problem

is normalized

by taking

^{R = 1}

and ₍₂ 1. Let D

= Be

ⁿ

(Rk).L, 1)

^{= n b+ = and ab+ _}

=

{~

^{E = 0 or}

Ixl

⁼

1}.

The intersection

pairing :

yields -+-1

^as intersection number for

appropriately

^chosen

generators

ⁱⁿ the

homology

groups above. Furthermore

naturality yields

^the

diagram

where j :

^{ab+ c} is inclusion. If g ^E

C(b+,

^{and G is as in the}

statement of the

lemma,

^then

j*

would be the trivial

homomorphism

^{and the}

intersection number would be

0,

^a contradiction.

These

preliminaries being completed,

we can now

give

^the

PROOF oF THEOREM 1.1. Assume for the moment that If ^c is not

a critical value of

I,

^{we can} invoke Lemma 1.3 with 9 ⁼

oc/2.

^{Let hEr be}

such that

218

Since

h) E C(Br

ⁿ

E)

^{and if}

I(w) 0,

^then

h(u))

⁼

u)

^{= u}

by 1)

of Lemma 1.3 and our choice

of e,

^{it follows that}

h )

^{e F. But}

then

by (1.5)

^and

3)

^{of Lemma}

1.3,

contrary

^to the definition of c.

To

complete

^the

proof,

^{we must show} c &#x3E; oe.

Using (12),

it suffices to show

h(B,.

^{r1 r1}

aBe

ⁿ

l# # 0

for all h E -P. Let K ⁼

h(Br

ⁿ

Ek+1).

^{Since K}

is

compact, by

^an

approximation

^{lemma of}

Leray-Schauder [9],

^{for all}

8 &#x3E;

0,

there exists a finite dimensional

subspace Fe

^c E and a

mapping is

^E

O(K,

^{such that}

11 i,,(u)

^{- for all u E K. We can assume}

Let

hE

⁼

ieoh.

Then

he

^E

C(B,.

ⁿ

Ek+1,

^{and -}

h(u)

^{as 8 - 0 uni-}

formly

^{for Observe that if}

by (1,)

and we have

Since E ⁼

Ek

EÐ

Ê,

^{u E E}

implies

^{u = v}

+

w, where ^v E

Ek,

^{I WE}

^Ê.

^We

can assume

+

^{Thus if u E}

(ôBr

^{r) U}

Ek ,

^{I it foi-}

lows that

Choosing e fl, identifying

^with

Rm, Ek, E,,,

^with

Rk,

^and

defining G(t, u)

⁼

the(u) + (1- t)u,

^it follows from

(1.7)-(1.8)

^that G satisfies the

hypotheses

^{of Lemma 1.4. Hence}

Ek+l)

ⁿ

aB, ^0.

^Conse-

quently

there exists uE

E Br

ⁿ

_Ek+1

such that E

aBe

ⁿ

^E.

^{As s -}

0, along

^a

subsequence

^we

^{have ue}

^{- u and}

Therefore and the

proof

^is

complete.

REMARK 1.9. If E is infinite

dimensional,

ⁱⁿ

general

^{I will not be bounded}

from above in contrast to the finite dimensional case where this is a con-

sequence of

(1,). By (I2), I

^{then has a}

positive

^{maximum c in}

Br.

^In

general

c c. However if c ⁼

Cy

I possesses

infinitely

many critical

points.

^In-

deed

(1.2)

^then shows I achieves its maximum on

h(Br

ⁿ

Ek+1)

^for

each h c- T

and this

implies

^{I has}

infinitely

many distinct critical

points corresponding

to c ⁼ c. Our next result

gives

another characterization of critical values of I in the finite case and a more

precise description

^{of the}

degenerate

^situation.

THEOREM 1.10. Let I E

C’(R-, R)

^and

satisfy (11)-(1s).

Then I possesses at least two

positive

^critical values characterized

by

and

where

Before

proving

^Theorem

1.10,

^{a few remarks are in} order. The notation

catma

^{refers to the}

Ljusternik-Schnirelman category

of the subset A of M.

In

(1.11)

^as admissible A we

only

consider A c M with A closed in Rm.

By

definition

cat~

^{A = 1} if A is contractible to a

point

^{in ~kl and}

cat,

A

= j if j

^{is the least}

integer

^{such that A can be covered}

by j

^{closed sets}

^{A with}

It is clear that any admissible A c M is

homotopic

to a subset of the unit

sphere

^{Sm-k in}

(Rk) 1.

^Moreover

catM

^{Sm-k = 2 since} if

eatm

^{8--k =}

1,

any

homotopy

^{of to a}

point

^{in ~} would induce a

homotopy

^{of Sm-k to a}

point

ⁱⁿ Sm-k. However as is well known

(see

^e.g.

[6]),

^{= 2 so no such}

homotopy

^{can exist.}

PROOF OF THEOREM 1.10. Define

Then _C1 ⁼ c since we can take A ⁼

{x}

^{for any x E M and C2 =}

^{b o c.}

^Note

further that

by (I2),

I &#x3E; 0 on

(2Sm-k

^{and therefore b} &#x3E; 0.

(Is) implies

^{I satis-}

fies

(PS)

^{on the set where} I&#x3E; 0. A

slightly strengthened

version of Lemma 1.3 and a standard

argument (similar

^{to the first}

paragraph

^{of the}

proof

^{of The-}

orem

1.1)

shows that b is a critical value of I and if c

= b, ^{cat~ .Kb}

^{= 2.}

(See

^e.g.

^{[6], [7],}

^or

[2]).

REMARK 1.13. We believe that b as defined in

(1.11) equals

^{c defined}

in

(1.2).

An examination of the

proof

of Theorem 1.10 shows that the

argument

in fact

gives

^the

following

result which

interchanges

the finite dimensional and finite codimensional

hypotheses (Ii)

^and

(Zg)

of Theorem 1.1.

THEOREM 1.14. Let E be a real Banach space, ^{I E}

01(E, R)

^and

satisfy (PS).

220

Suppose

^{that E =}

^Ek

^EÐ

^E

^where

7~ ~ 1, Ek

^{is and I}

satisfies

There are constants

(16)

^{1 is bounded}

f rom

^above.

Then I possesses ^at least two

positive

critical values characterized

by

where A is

compact in

^{E and M =}

EB-P.

^Moreover

if b,

⁼

b2

⁼

d,

PROOF. Immediate from that of

Theorem

^1.10 and the remark _pre-

ceding

^it.

2. -

Applications

^to

partial

differential

equations.

In this section we shall show how Theorems 1.1 and 1.14 can be

employed

to obtain existence theorems for semilinear

elliptic boundary

^value

problems.

Consider

where S~ is a bounded domain in Rn with ^a smooth

boundary, Z

^is

uniformly elliptic

ⁱⁿ

D

with smooth

coefficients, ^{c ~ 0}

^{in is} smooth and

positive

in

S~,

and p is smooth and satisfies

There are constants .lVl &#x3E; 0 and 6 E

(0, i)

^{such that}

If

n 2, (p1 )

^{can be}

considerably improved.

Set

Formally

^critical

points

^{of I in E = are} weak solutions of

(2.1).

The smoothness of

L, a, p,

^and

(PI)

and standard

regularity

results then

imply

such weak solutions are smooth functions. Thus we focus our atten- tion on

finding

^critical

points

^{of I in E.}

Consider the linear

eigenvalue problem

As is well

known, (2.3)

possesses ^an unbounded sequence of

eigenvalues

0

ÂI ~2 c

^...

Am -

^{oo as m -~ oo with}

corresponding eigenfunctions

VIL 7 ... 9 V, 7 .... It ^was shown in

[1 ]

that if p satisfies and

~,1

&#x3E;

1, (2.1)

possesses ^a

positive

^solution

(i.e. u

&#x3E; 0 in

S~)

^{and a}

negative

^solution.

If the

argument

^of

[1]

fails unless

p(x, z)

^{is odd in z.} We will show:

THEOREM 2.4.

I f

p

satisfies (p,)-(p,)

^and

then

(2.1)

possesses ^a nontrivial solution.

PROOF. Because of the result

just mentioned,

^{we can assume}

In fact suppose

,k c 1 +1.

^Let

Ek

⁼

span{vl, ... , vk

^and

^Ê E-L

^the

orthogonal complement

^of

Ek. By (P4)’ I ~ 0

^on

Ek .

^{It was} further shown in

[1]

^that

(pl)-(p3) imply

^{that I E}

C’(E, R)

and satisfies

(1,),

^and

(PS).

Hence Theorem 2.4 follows

immediately

^{from Theorem 1.1.}

REMARK 2.5. As was noted above

for A, &#x3E; 1, (2.1)

^{has a}

positive

^{and a}

negative

solution. In contrast we next show:

COROLLARY 2.6. Tlnder the

hypotheses of

^Theorem

2.4, if ~,1 c 1, (2.1)

^does

not have a

positive (or negative)

^solution

u(x)

^unless

Â1

⁼

1, u(x)

^{is a}

multiple of vl(x)

^and

p (x, u(x))

^{== 0.}

PROOF.

Suppose u

^{is a}

positive

solution of

(2.1)

^{with Then.}

222

Since we can assume

which is

impossibile

^unless

p(x, z)

^{=- 0 for and in}

which case ’U =

f3v1 .

Next we illustrate how Theorem 1.14 can be used in similar situations.

Consider

(2.1) again

^where p satisfies _{and e.g.}

No

growth

conditions are needed for this case since

(p~) implies

^{that az}

-f -

p

can be redefined to be

independent

^{of z for}

large Izl so

the modified

nonlinearity

is

uniformly

bounded. Moreover solutions of the modified

equation

^{are still}

solutions of

(2.1). (See

e.g. Theorem 3.4 of

[2]).

^{It is}

easily

^{verified that}

if

~,k C 1 ~ ~,k+~ ,

¹

Ek , ^~

^{are as in Theorem 2.4 and}

where P

is the

primative

of the modification of az

-E-

p, then J satisfies the

hypotheses

^{of Theorem 1.14.}

Since

stronger

^{results can be obtained for this}

problem using

^methods based on

Leray-Schauder degree theory [8]

^{we will not} carry ^out the details here. However if L were

replaced by

^a

higher

^order

divergence

^structure

elliptic operator

^one could obtain results

using

Theorem 1.14 where

degree

theoretic methods would fail.

REFERENCES

[1]

A. AMBROSETTI - P. H. RABINOWITZ, Dual variational methods in critical

point theory

^and

applications,

J. Functional Anal., 14

(1973),

pp. 349-381.

[2]

^{P. H.} RABINOWITZ, Variational methods

for

^nonlinear

eigenvalue problems, Eigenvalues of

^{Nonlinear Problems, Edizioni} Cremonese, Rome

(1974),

pp. 141-195.

[3]

^{D. C.} CLARK, ^{A variant}

of

the Lusternik-Schnirelman

theory,

Ind. Univ. Math. J., 22

(1972),

pp. ^65-74.

[4]

^A. DOLD, ^{Lectures on}

Algebraic Topology, Springer-Verlag,

^{Berlin, 1972.}

[5] ^{P. H.} RABINOWITZ, Free vibrations

for

^a semitinear wave

equation,

^to appear Comm. Pure

Appl.

^Math.

[6] ^{J. T.} SCHWARTZ, Nonlinear Functional

Analysis,

^{lecture notes,} Courant Inst.

of Math. Sc., New York Univ., ^1965.

[7] ^{R. S.} PALAIS, Critical

point theory

and the minimax

principle,

^Proc.

Sym.

^Pure Math.,

15,

^A.M.S. Providence, ^R. I., (1970), pp. ^185-212.

[8] ^{P. H.} RABINOWITZ, Some aspects

of

^nonlinear

eigenvalue problems, Rocky

^Mtn.

Math. J., 3 (1973), pp. ^161-202.

[9]

J. LERAY - J. SCHAUDER,

Topologie

^et

equations fonctionnelles,

Ann. Sci. Ecole Norm.

Sup.,

3, ⁵¹ (1934), pp. ^45-78.